Today's integrated circuits (ICs) contain features which are not easily resolved by available lithography tools, so proper printing of critical feature dimensions (CDs) requires that compensating adjustments be made to the IC shapes deployed on the mask. In so-called model-based optical-proximity-correction (MBOPC), the appropriate adjustments are determined by simulating the lithographic process, and in particular, providing a means to simulate the image at the wafer plane. Ultimately, a determination of the image formed in the resist layer (e.g. the latent resist image) is desired, but often the aerial image at the wafer plane is used as an approximation for the latent image in the resist.
Conventional image simulation is typically done using the Hopkins integral for scalar partial coherent image formation, where the expression for the aerial image intensity I0 is given by,I0({right arrow over (r)})=∫∫∫∫d{right arrow over (r)}′d{right arrow over (r)}″h({right arrow over (r)}−{right arrow over (r)}′) h*({right arrow over (r)}−{right arrow over (r)}″)m({right arrow over (r)}′)m*({right arrow over (r)}″),  Equation 1where,                h is the lens impulse response function (also known as the point spread function or PSF);        j is the coherence;        m is the mask transmission function;        * indicates the complex conjugate; and        {right arrow over (r)} is the position of the image.The integration is typically performed over the mask. The expression:h({right arrow over (r)}−{right arrow over (r)}′)h*({right arrow over (r)}−{right arrow over (r)}′)j({right arrow over (r)}′−{right arrow over (r)}″)  Equation 2is known as the Hopkins kernel, which is a fixed two-dimensional (2D) function for a given system.        
This 4-dimensional (4D) Hopkins integral (Equation 1) may be approximated as an incoherent sum of 2-dimensional (2D) coherent image integrals. This method of computing the Hopkins integral is known as the sum of coherent systems (SOCS) procedure. In the SOCS procedure, an optimal n-term approximation to the partially coherent Hopkins integral is:
                                                        I              0                        ⁡                          (                              r                →                            )                                ≅                                    ∑                              k                =                1                            n                        ⁢                                          λ                k                            ⁢                                                                                                            (                                              m                        ⊗                                                  ϕ                          k                                                                    )                                        ⁢                                          (                                              r                        →                                            )                                                                                        2                                                    ,                            Equation        ⁢                                  ⁢        3            where {circle around (×)} represents the two-dimensional (2D) convolution operation, λk, φk({right arrow over (r)}) represent the kth eigenvalue and eigenfunction of the Hopkins kernel, respectively, derived from the Mercer expansion of:
                                                        h              ⁡                              (                                                      r                    →                                    ′                                )                                      ⁢                                          h                *                            ⁡                              (                                                      r                    →                                    ″                                )                                      ⁢                          j              ⁡                              (                                                                            r                      →                                        ′                                    -                                                            r                      →                                        ″                                                  )                                              =                                    ∑                              k                =                1                            ∞                        ⁢                                          λ                k                            ⁢                                                ϕ                  k                                ⁡                                  (                                                            r                      →                                        ′                                    )                                            ⁢                                                ϕ                  k                                ⁡                                  (                                                            r                      →                                        ″                                    )                                                                    ,                            Equation        ⁢                                  ⁢        4            which suggests that a partially coherent imaging problem can be optimally approximated by a finite sum of coherent images obtained, for example, by linear convolution. Typically, the source and the mask polygons are decomposed (e.g. into grids or sectors), and each field image is computed as an incoherent sum of coherent sub-images (also referred to as component-images, or pre-images). The total intensity at an image point {right arrow over (r)} in question is then the sum over all component images. In the SOCS approximation, the number of coherent sub-images that must be calculated is minimized, for example, by diagonalizing the image matrix to achieve an acceptable approximate matrix of minimal rank by eigenvalue decomposition. For example, even a large-fill source can be adequately approximated when the number of 2D convolutions n is about 10.
The mask transmission function, m, which can be approximated by a binary mask design pattern, usually consisting of polygons of one or more transmission, can be represented in different ways, such as grid cells or a sector decomposition, and the image at a point {right arrow over (r)} may be represented by a finite incoherent sum of weighted coherent convolutions of the mask transmission function m and the eigenfunctions φk. Each of the eigenfunctions, and its convolution with a possible mask sector, may be pre-computed, thus providing a fast method of computing the aerial image I0. The aerial image is often assumed to be an adequate approximation of the resist latent image. To account for resist effects, previous methods have used post processing the aerial image by applying a lumped parameter model or by applying resist blurring. However, in extending model-based-optical-proximity-correction (MBOPC) to the sub-100 nm dimensions of next-generation IC products, the prior art has a number of limitations.
For example, the scalar treatment is applicable for numerical apertures (NAs) less than about 0.7, where the angle between interfering orders is fairly small in resist, so that the electric fields in different beams are almost perpendicular or anti-perpendicular when they interfere. Under these circumstances, superposition is almost scalar. However, numerical apertures (NAs) of at least 0.85 must be employed when extending optical lithography to the sub-100 nm dimensions. At the resulting steep obliquities the standard scalar Hopkins integral becomes inaccurate, and the vector character of the electric field must be considered. It is known that this can be accomplished by calculating independent images in each of the global Cartesian coordinates of the electric field (Ex, Ey, Ez), and then summing these images over each of the allowed orientations of the illuminating polarization, and again over each of the different source directions that illuminate the mask. However, the computational efficiency of this procedure is not adequate for MBOPC.
Adam et al. (K. Adam, Y. Granik, A. Torres and N. B. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” in SPIE vol. 5040, Optical Microlithography XVI, ed. Anthony Yen (2003), p. 78–91) and Hafeman et al. (in SPIE vol. 5040, Optical Microlithography XVI, ed. Anthony Yen (2003), p. 700) disclosed an extension of the scalar Hopkins imaging equation to include vectorial addition of the electromagnetic (EM) field inside a film. However, the approaches of Adam et al. and Hafeman et al. ignore the z-component of the field, and do not take into account effects such as lens birefringence, tailored source polarization, or blur from the resist or mask.
The scalar treatment of image formation ignores illumination polarization, and assumes that the lens and resist surfaces introduce negligible partial polarization. However, OPC for future lithography will also have to take into account the polarization properties of the lens itself. As NA and lens complexity increases, the polarization state of the output beam is changed by cumulative polarization-dependent reflection losses that arise at the surfaces of the lens elements. Attainable performance in the antireflection coatings that inhibit these losses will become increasingly poor as wavelength decreases into the deep ultraviolet (UV). Forthcoming 157 nm lenses will be birefringent even in their bulk substrates, due to spatial dispersion in the element substrates. Skew incidence at beamsplitter and other coatings will distort the light polarization. There is also interest in sources that deliberately introduce polarization variations between rays in order to minimize the contrast loss that arises in transverse magnetic (TM) polarization at high NA (i.e. polarization-tailored sources). The exposed image is also impacted by the resist film stack, due to multiple polarization-dependent reflections between the various interfaces. The anti-reflective (AR) films that inhibit these reflections can be less effective over the broad angular ranges that arise at high NA. Refraction at the top surface of the resist gives rise to spherical aberration in the transmitted image.
Finally, minimum feature sizes in next generation ICs are beginning to approach the resolution of the photoresist, and this must be taken into account during MBOPC.
Methods are known for modeling each of these phenomena over mask areas of modest size. Vector images can be calculated by summing over all illumination and image-plane polarizations, and over all source points.
Resist blur may occur due to the finite resolution of the resist. Propagation in spatially dispersive media has been treated in the physics literature, and this analysis has now been applied to 157 nm lithographic lenses. Finite resist resolution can be treated in an approximate way by blurring the exposed optical image using a resist kernel, which in the frequency domain is equivalent to frequency filtering the 4D Hopkins integral. It is known that a post-exposure blurring in the resist arises in the chemical image that is acted on by the resist developer; this resist blurring can be accounted for by convolution of the optical image with a blur function, or equivalently by attenuation of the image spatial frequency content by a modulation transfer function (for example, see J. Garofalo et al., “Reduction of ASIC Gate-Level line-end shortening by Mask Compensation,” in SPIE v.2440—Optical/Laser Microlithography VIII, ed. Timothy A. Brunner (SPIE, 1995), p. 171.). Hoffnagle et al. (in “Method of measuring the spatial resolution of a photoresist,” Optics Letters 27, no.20 (2002): p. 1776.) have shown that the resist modulation transfer function (also known as the effective resist MTF) can be determined for a particular spatial frequency from critical dimension (CD) measurements in resist that has been exposed to a pair of interfering sine waves (this interference pattern exhibiting essentially 100% optical contrast). The latent image contrast deduced in this way can be substantially less than 100%, and this modulation loss becomes steadily more important as feature-sizes come closer to the resolution limit of resists. For example, at a pitch of 225 nm, the UV2 resist analyzed in Hoffnagle et al. transfers to the latent chemical image only about 50% of the modulation in the exposing optical image. More modest contrast losses arise even at relatively coarse spatial frequencies, and can give rise to proximity effects of relatively long range.
However, such methods to account for resist blurring involve a direct spatial domain convolution of the resist blur function with the continuous optical image, which is relatively slow to compute, as compared to the polygons of the kind that are deployed on the mask, which are relatively simple to handle, and can be relatively fast to compute. Unfortunately, future MBOPC will only be practical if a way can be found to calculate these effects very quickly.
Other aspects of the MBOPC process (e.g. etch simulation, simulation of mask electromagnetic effects, polygon edge adjustment, as well as many non-lithographic issues) are not directly considered in the present discussion; however, it is important to bear in mind that lithographic simulation is only one small aspect of the full IC design process. Mask polygons must be stored in a recognized data structure of complex hierarchical organization; the patterns are arrived at after considerable effort that involves a long succession of circuit and device design software. Additional processing software converts the post-OPC shapes to a format used in mask writing. Because of the complexity of this computer-aided design (CAD) process, it has become customary to use suites of software design tools (including tools for MBOPC) to avoid problems when migrating the chip data from one stage of the design process to the next. Improved MBOPC methods must be compatible with this design flow. It is also desirable that improved methods for lithographic simulation be compatible with existing OPC programs.
The SOCS method gives a fast and accurate enough algorithm for computing the scalar aerial image (AI). This would be enough to satisfy and even exceed any practical requirements if only the AI was needed as the final output of the algorithm. However, because of the presence of the resist development step, further computation is needed to get the resist image from the AI.
Images obtained using the SOCS method are calculated by pre-storing the corresponding images of all possible semi-infinite sectors from which the mask polygons are composed. The images of polygonal mask features can therefore be calculated very rapidly.
These pre-stored tables are based only on the scalar Hopkins integral. To take into account such phenomena as resist resolution, vector imaging, resist thin-film effects, illumination polarization, and lens birefringence, we need to obtain new tables that are able to reproduce the effects of these phenomena as if the phenomena arose from an incoherent sum of coherent images.
Accordingly, there is a need for an efficient method of simulating images that more accurately includes non-scalar effects (i.e. “non-Hopkins” effects) including the vector electric field, polarization effects, and is applicable for computing a variety of images including an aerial image or a resist image. In addition, it would be desirable to implement such a method that can be incorporated into existing computer codes without significant restructuring of the code.